Properties

Label 752.4.a.c.1.2
Level $752$
Weight $4$
Character 752.1
Self dual yes
Analytic conductor $44.369$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [752,4,Mod(1,752)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(752, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("752.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 752 = 2^{4} \cdot 47 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 752.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.3694363243\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 47)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.68740\) of defining polynomial
Character \(\chi\) \(=\) 752.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.777884 q^{3} +9.19383 q^{5} +30.3255 q^{7} -26.3949 q^{9} +O(q^{10})\) \(q+0.777884 q^{3} +9.19383 q^{5} +30.3255 q^{7} -26.3949 q^{9} -22.4298 q^{11} -62.0257 q^{13} +7.15173 q^{15} -72.1639 q^{17} +25.0550 q^{19} +23.5897 q^{21} -103.176 q^{23} -40.4735 q^{25} -41.5350 q^{27} -234.381 q^{29} -198.714 q^{31} -17.4477 q^{33} +278.807 q^{35} -203.083 q^{37} -48.2488 q^{39} +210.889 q^{41} -111.430 q^{43} -242.670 q^{45} -47.0000 q^{47} +576.634 q^{49} -56.1351 q^{51} +499.576 q^{53} -206.215 q^{55} +19.4898 q^{57} +562.752 q^{59} +548.091 q^{61} -800.437 q^{63} -570.254 q^{65} +760.831 q^{67} -80.2586 q^{69} +668.059 q^{71} -1145.92 q^{73} -31.4836 q^{75} -680.193 q^{77} -975.010 q^{79} +680.353 q^{81} -698.827 q^{83} -663.463 q^{85} -182.321 q^{87} +451.477 q^{89} -1880.96 q^{91} -154.577 q^{93} +230.351 q^{95} -390.906 q^{97} +592.031 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 5 q^{3} - 6 q^{5} + 45 q^{7} - 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 5 q^{3} - 6 q^{5} + 45 q^{7} - 42 q^{9} - 2 q^{11} - 80 q^{13} - 14 q^{15} - 39 q^{17} + 24 q^{19} + 24 q^{21} - 120 q^{23} - 171 q^{25} - 64 q^{27} - 184 q^{29} + 4 q^{31} - 208 q^{33} + 156 q^{35} - 589 q^{37} - 60 q^{39} - 92 q^{41} + 250 q^{43} - 78 q^{45} - 141 q^{47} + 30 q^{49} - 317 q^{51} + 459 q^{53} - 448 q^{55} + 216 q^{57} - 579 q^{59} + 267 q^{61} - 1044 q^{63} - 424 q^{65} + 540 q^{67} + 642 q^{69} - 749 q^{71} - 1924 q^{73} - 473 q^{75} - 288 q^{77} - 805 q^{79} + 291 q^{81} - 712 q^{83} - 1038 q^{85} - 1216 q^{87} + 835 q^{89} - 2040 q^{91} - 1500 q^{93} + 312 q^{95} - 2243 q^{97} - 554 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.777884 0.149704 0.0748519 0.997195i \(-0.476152\pi\)
0.0748519 + 0.997195i \(0.476152\pi\)
\(4\) 0 0
\(5\) 9.19383 0.822321 0.411161 0.911563i \(-0.365124\pi\)
0.411161 + 0.911563i \(0.365124\pi\)
\(6\) 0 0
\(7\) 30.3255 1.63742 0.818711 0.574206i \(-0.194689\pi\)
0.818711 + 0.574206i \(0.194689\pi\)
\(8\) 0 0
\(9\) −26.3949 −0.977589
\(10\) 0 0
\(11\) −22.4298 −0.614803 −0.307401 0.951580i \(-0.599459\pi\)
−0.307401 + 0.951580i \(0.599459\pi\)
\(12\) 0 0
\(13\) −62.0257 −1.32330 −0.661648 0.749815i \(-0.730143\pi\)
−0.661648 + 0.749815i \(0.730143\pi\)
\(14\) 0 0
\(15\) 7.15173 0.123105
\(16\) 0 0
\(17\) −72.1639 −1.02955 −0.514774 0.857326i \(-0.672124\pi\)
−0.514774 + 0.857326i \(0.672124\pi\)
\(18\) 0 0
\(19\) 25.0550 0.302526 0.151263 0.988494i \(-0.451666\pi\)
0.151263 + 0.988494i \(0.451666\pi\)
\(20\) 0 0
\(21\) 23.5897 0.245128
\(22\) 0 0
\(23\) −103.176 −0.935373 −0.467687 0.883894i \(-0.654912\pi\)
−0.467687 + 0.883894i \(0.654912\pi\)
\(24\) 0 0
\(25\) −40.4735 −0.323788
\(26\) 0 0
\(27\) −41.5350 −0.296052
\(28\) 0 0
\(29\) −234.381 −1.50081 −0.750405 0.660978i \(-0.770142\pi\)
−0.750405 + 0.660978i \(0.770142\pi\)
\(30\) 0 0
\(31\) −198.714 −1.15129 −0.575647 0.817698i \(-0.695250\pi\)
−0.575647 + 0.817698i \(0.695250\pi\)
\(32\) 0 0
\(33\) −17.4477 −0.0920383
\(34\) 0 0
\(35\) 278.807 1.34649
\(36\) 0 0
\(37\) −203.083 −0.902343 −0.451171 0.892437i \(-0.648994\pi\)
−0.451171 + 0.892437i \(0.648994\pi\)
\(38\) 0 0
\(39\) −48.2488 −0.198102
\(40\) 0 0
\(41\) 210.889 0.803303 0.401651 0.915793i \(-0.368436\pi\)
0.401651 + 0.915793i \(0.368436\pi\)
\(42\) 0 0
\(43\) −111.430 −0.395184 −0.197592 0.980284i \(-0.563312\pi\)
−0.197592 + 0.980284i \(0.563312\pi\)
\(44\) 0 0
\(45\) −242.670 −0.803892
\(46\) 0 0
\(47\) −47.0000 −0.145865
\(48\) 0 0
\(49\) 576.634 1.68115
\(50\) 0 0
\(51\) −56.1351 −0.154127
\(52\) 0 0
\(53\) 499.576 1.29476 0.647378 0.762169i \(-0.275866\pi\)
0.647378 + 0.762169i \(0.275866\pi\)
\(54\) 0 0
\(55\) −206.215 −0.505565
\(56\) 0 0
\(57\) 19.4898 0.0452894
\(58\) 0 0
\(59\) 562.752 1.24176 0.620882 0.783904i \(-0.286775\pi\)
0.620882 + 0.783904i \(0.286775\pi\)
\(60\) 0 0
\(61\) 548.091 1.15042 0.575212 0.818004i \(-0.304919\pi\)
0.575212 + 0.818004i \(0.304919\pi\)
\(62\) 0 0
\(63\) −800.437 −1.60072
\(64\) 0 0
\(65\) −570.254 −1.08817
\(66\) 0 0
\(67\) 760.831 1.38732 0.693659 0.720304i \(-0.255998\pi\)
0.693659 + 0.720304i \(0.255998\pi\)
\(68\) 0 0
\(69\) −80.2586 −0.140029
\(70\) 0 0
\(71\) 668.059 1.11668 0.558338 0.829613i \(-0.311439\pi\)
0.558338 + 0.829613i \(0.311439\pi\)
\(72\) 0 0
\(73\) −1145.92 −1.83726 −0.918629 0.395121i \(-0.870703\pi\)
−0.918629 + 0.395121i \(0.870703\pi\)
\(74\) 0 0
\(75\) −31.4836 −0.0484722
\(76\) 0 0
\(77\) −680.193 −1.00669
\(78\) 0 0
\(79\) −975.010 −1.38857 −0.694286 0.719699i \(-0.744280\pi\)
−0.694286 + 0.719699i \(0.744280\pi\)
\(80\) 0 0
\(81\) 680.353 0.933269
\(82\) 0 0
\(83\) −698.827 −0.924171 −0.462086 0.886835i \(-0.652899\pi\)
−0.462086 + 0.886835i \(0.652899\pi\)
\(84\) 0 0
\(85\) −663.463 −0.846620
\(86\) 0 0
\(87\) −182.321 −0.224677
\(88\) 0 0
\(89\) 451.477 0.537713 0.268857 0.963180i \(-0.413354\pi\)
0.268857 + 0.963180i \(0.413354\pi\)
\(90\) 0 0
\(91\) −1880.96 −2.16679
\(92\) 0 0
\(93\) −154.577 −0.172353
\(94\) 0 0
\(95\) 230.351 0.248774
\(96\) 0 0
\(97\) −390.906 −0.409180 −0.204590 0.978848i \(-0.565586\pi\)
−0.204590 + 0.978848i \(0.565586\pi\)
\(98\) 0 0
\(99\) 592.031 0.601024
\(100\) 0 0
\(101\) −1112.66 −1.09618 −0.548089 0.836420i \(-0.684644\pi\)
−0.548089 + 0.836420i \(0.684644\pi\)
\(102\) 0 0
\(103\) 882.574 0.844298 0.422149 0.906527i \(-0.361276\pi\)
0.422149 + 0.906527i \(0.361276\pi\)
\(104\) 0 0
\(105\) 216.880 0.201574
\(106\) 0 0
\(107\) −840.742 −0.759604 −0.379802 0.925068i \(-0.624008\pi\)
−0.379802 + 0.925068i \(0.624008\pi\)
\(108\) 0 0
\(109\) −1321.06 −1.16086 −0.580432 0.814309i \(-0.697116\pi\)
−0.580432 + 0.814309i \(0.697116\pi\)
\(110\) 0 0
\(111\) −157.975 −0.135084
\(112\) 0 0
\(113\) 701.293 0.583824 0.291912 0.956445i \(-0.405709\pi\)
0.291912 + 0.956445i \(0.405709\pi\)
\(114\) 0 0
\(115\) −948.579 −0.769177
\(116\) 0 0
\(117\) 1637.16 1.29364
\(118\) 0 0
\(119\) −2188.40 −1.68580
\(120\) 0 0
\(121\) −827.906 −0.622018
\(122\) 0 0
\(123\) 164.047 0.120257
\(124\) 0 0
\(125\) −1521.34 −1.08858
\(126\) 0 0
\(127\) −1142.65 −0.798374 −0.399187 0.916869i \(-0.630708\pi\)
−0.399187 + 0.916869i \(0.630708\pi\)
\(128\) 0 0
\(129\) −86.6795 −0.0591605
\(130\) 0 0
\(131\) −104.451 −0.0696639 −0.0348319 0.999393i \(-0.511090\pi\)
−0.0348319 + 0.999393i \(0.511090\pi\)
\(132\) 0 0
\(133\) 759.803 0.495363
\(134\) 0 0
\(135\) −381.866 −0.243450
\(136\) 0 0
\(137\) −543.914 −0.339195 −0.169598 0.985513i \(-0.554247\pi\)
−0.169598 + 0.985513i \(0.554247\pi\)
\(138\) 0 0
\(139\) 41.8374 0.0255295 0.0127647 0.999919i \(-0.495937\pi\)
0.0127647 + 0.999919i \(0.495937\pi\)
\(140\) 0 0
\(141\) −36.5605 −0.0218365
\(142\) 0 0
\(143\) 1391.22 0.813565
\(144\) 0 0
\(145\) −2154.86 −1.23415
\(146\) 0 0
\(147\) 448.554 0.251674
\(148\) 0 0
\(149\) 790.139 0.434435 0.217217 0.976123i \(-0.430302\pi\)
0.217217 + 0.976123i \(0.430302\pi\)
\(150\) 0 0
\(151\) 2073.79 1.11763 0.558816 0.829292i \(-0.311256\pi\)
0.558816 + 0.829292i \(0.311256\pi\)
\(152\) 0 0
\(153\) 1904.76 1.00648
\(154\) 0 0
\(155\) −1826.95 −0.946734
\(156\) 0 0
\(157\) 2501.12 1.27141 0.635704 0.771933i \(-0.280710\pi\)
0.635704 + 0.771933i \(0.280710\pi\)
\(158\) 0 0
\(159\) 388.612 0.193830
\(160\) 0 0
\(161\) −3128.85 −1.53160
\(162\) 0 0
\(163\) 97.0569 0.0466386 0.0233193 0.999728i \(-0.492577\pi\)
0.0233193 + 0.999728i \(0.492577\pi\)
\(164\) 0 0
\(165\) −160.412 −0.0756850
\(166\) 0 0
\(167\) 1826.71 0.846437 0.423218 0.906028i \(-0.360900\pi\)
0.423218 + 0.906028i \(0.360900\pi\)
\(168\) 0 0
\(169\) 1650.19 0.751111
\(170\) 0 0
\(171\) −661.323 −0.295746
\(172\) 0 0
\(173\) −1843.63 −0.810223 −0.405111 0.914267i \(-0.632767\pi\)
−0.405111 + 0.914267i \(0.632767\pi\)
\(174\) 0 0
\(175\) −1227.38 −0.530177
\(176\) 0 0
\(177\) 437.756 0.185897
\(178\) 0 0
\(179\) 370.515 0.154713 0.0773565 0.997003i \(-0.475352\pi\)
0.0773565 + 0.997003i \(0.475352\pi\)
\(180\) 0 0
\(181\) −866.586 −0.355872 −0.177936 0.984042i \(-0.556942\pi\)
−0.177936 + 0.984042i \(0.556942\pi\)
\(182\) 0 0
\(183\) 426.351 0.172223
\(184\) 0 0
\(185\) −1867.11 −0.742016
\(186\) 0 0
\(187\) 1618.62 0.632969
\(188\) 0 0
\(189\) −1259.57 −0.484763
\(190\) 0 0
\(191\) 3667.49 1.38937 0.694687 0.719312i \(-0.255543\pi\)
0.694687 + 0.719312i \(0.255543\pi\)
\(192\) 0 0
\(193\) −4676.91 −1.74431 −0.872154 0.489231i \(-0.837277\pi\)
−0.872154 + 0.489231i \(0.837277\pi\)
\(194\) 0 0
\(195\) −443.591 −0.162904
\(196\) 0 0
\(197\) 3700.99 1.33850 0.669251 0.743037i \(-0.266615\pi\)
0.669251 + 0.743037i \(0.266615\pi\)
\(198\) 0 0
\(199\) 1361.63 0.485042 0.242521 0.970146i \(-0.422026\pi\)
0.242521 + 0.970146i \(0.422026\pi\)
\(200\) 0 0
\(201\) 591.838 0.207687
\(202\) 0 0
\(203\) −7107.72 −2.45746
\(204\) 0 0
\(205\) 1938.88 0.660573
\(206\) 0 0
\(207\) 2723.31 0.914411
\(208\) 0 0
\(209\) −561.977 −0.185994
\(210\) 0 0
\(211\) 2873.21 0.937442 0.468721 0.883346i \(-0.344715\pi\)
0.468721 + 0.883346i \(0.344715\pi\)
\(212\) 0 0
\(213\) 519.672 0.167171
\(214\) 0 0
\(215\) −1024.47 −0.324968
\(216\) 0 0
\(217\) −6026.10 −1.88515
\(218\) 0 0
\(219\) −891.393 −0.275044
\(220\) 0 0
\(221\) 4476.02 1.36240
\(222\) 0 0
\(223\) −3356.14 −1.00782 −0.503909 0.863757i \(-0.668105\pi\)
−0.503909 + 0.863757i \(0.668105\pi\)
\(224\) 0 0
\(225\) 1068.29 0.316531
\(226\) 0 0
\(227\) 3991.01 1.16693 0.583464 0.812139i \(-0.301697\pi\)
0.583464 + 0.812139i \(0.301697\pi\)
\(228\) 0 0
\(229\) −968.028 −0.279341 −0.139670 0.990198i \(-0.544604\pi\)
−0.139670 + 0.990198i \(0.544604\pi\)
\(230\) 0 0
\(231\) −529.111 −0.150705
\(232\) 0 0
\(233\) −1227.70 −0.345189 −0.172595 0.984993i \(-0.555215\pi\)
−0.172595 + 0.984993i \(0.555215\pi\)
\(234\) 0 0
\(235\) −432.110 −0.119948
\(236\) 0 0
\(237\) −758.444 −0.207875
\(238\) 0 0
\(239\) −647.534 −0.175253 −0.0876265 0.996153i \(-0.527928\pi\)
−0.0876265 + 0.996153i \(0.527928\pi\)
\(240\) 0 0
\(241\) −4174.42 −1.11576 −0.557880 0.829922i \(-0.688385\pi\)
−0.557880 + 0.829922i \(0.688385\pi\)
\(242\) 0 0
\(243\) 1650.68 0.435766
\(244\) 0 0
\(245\) 5301.47 1.38244
\(246\) 0 0
\(247\) −1554.05 −0.400332
\(248\) 0 0
\(249\) −543.606 −0.138352
\(250\) 0 0
\(251\) 1368.39 0.344112 0.172056 0.985087i \(-0.444959\pi\)
0.172056 + 0.985087i \(0.444959\pi\)
\(252\) 0 0
\(253\) 2314.20 0.575070
\(254\) 0 0
\(255\) −516.097 −0.126742
\(256\) 0 0
\(257\) 1709.94 0.415031 0.207515 0.978232i \(-0.433462\pi\)
0.207515 + 0.978232i \(0.433462\pi\)
\(258\) 0 0
\(259\) −6158.59 −1.47752
\(260\) 0 0
\(261\) 6186.47 1.46718
\(262\) 0 0
\(263\) −5368.95 −1.25880 −0.629399 0.777082i \(-0.716699\pi\)
−0.629399 + 0.777082i \(0.716699\pi\)
\(264\) 0 0
\(265\) 4593.02 1.06471
\(266\) 0 0
\(267\) 351.197 0.0804977
\(268\) 0 0
\(269\) −4213.70 −0.955070 −0.477535 0.878613i \(-0.658470\pi\)
−0.477535 + 0.878613i \(0.658470\pi\)
\(270\) 0 0
\(271\) −4769.83 −1.06918 −0.534588 0.845113i \(-0.679533\pi\)
−0.534588 + 0.845113i \(0.679533\pi\)
\(272\) 0 0
\(273\) −1463.17 −0.324377
\(274\) 0 0
\(275\) 907.810 0.199066
\(276\) 0 0
\(277\) 5651.38 1.22584 0.612922 0.790144i \(-0.289994\pi\)
0.612922 + 0.790144i \(0.289994\pi\)
\(278\) 0 0
\(279\) 5245.04 1.12549
\(280\) 0 0
\(281\) 3264.83 0.693109 0.346554 0.938030i \(-0.387352\pi\)
0.346554 + 0.938030i \(0.387352\pi\)
\(282\) 0 0
\(283\) −2019.38 −0.424169 −0.212084 0.977251i \(-0.568025\pi\)
−0.212084 + 0.977251i \(0.568025\pi\)
\(284\) 0 0
\(285\) 179.186 0.0372424
\(286\) 0 0
\(287\) 6395.32 1.31534
\(288\) 0 0
\(289\) 294.634 0.0599703
\(290\) 0 0
\(291\) −304.079 −0.0612558
\(292\) 0 0
\(293\) 3432.28 0.684354 0.342177 0.939636i \(-0.388836\pi\)
0.342177 + 0.939636i \(0.388836\pi\)
\(294\) 0 0
\(295\) 5173.85 1.02113
\(296\) 0 0
\(297\) 931.621 0.182014
\(298\) 0 0
\(299\) 6399.54 1.23778
\(300\) 0 0
\(301\) −3379.16 −0.647082
\(302\) 0 0
\(303\) −865.521 −0.164102
\(304\) 0 0
\(305\) 5039.06 0.946019
\(306\) 0 0
\(307\) 6327.92 1.17640 0.588198 0.808717i \(-0.299838\pi\)
0.588198 + 0.808717i \(0.299838\pi\)
\(308\) 0 0
\(309\) 686.540 0.126395
\(310\) 0 0
\(311\) −9118.69 −1.66262 −0.831308 0.555812i \(-0.812408\pi\)
−0.831308 + 0.555812i \(0.812408\pi\)
\(312\) 0 0
\(313\) 4611.70 0.832808 0.416404 0.909180i \(-0.363290\pi\)
0.416404 + 0.909180i \(0.363290\pi\)
\(314\) 0 0
\(315\) −7359.09 −1.31631
\(316\) 0 0
\(317\) −4243.78 −0.751907 −0.375954 0.926638i \(-0.622685\pi\)
−0.375954 + 0.926638i \(0.622685\pi\)
\(318\) 0 0
\(319\) 5257.12 0.922702
\(320\) 0 0
\(321\) −654.000 −0.113716
\(322\) 0 0
\(323\) −1808.06 −0.311466
\(324\) 0 0
\(325\) 2510.40 0.428467
\(326\) 0 0
\(327\) −1027.63 −0.173786
\(328\) 0 0
\(329\) −1425.30 −0.238842
\(330\) 0 0
\(331\) 3803.59 0.631613 0.315807 0.948824i \(-0.397725\pi\)
0.315807 + 0.948824i \(0.397725\pi\)
\(332\) 0 0
\(333\) 5360.36 0.882120
\(334\) 0 0
\(335\) 6994.95 1.14082
\(336\) 0 0
\(337\) −6419.12 −1.03760 −0.518801 0.854895i \(-0.673621\pi\)
−0.518801 + 0.854895i \(0.673621\pi\)
\(338\) 0 0
\(339\) 545.524 0.0874006
\(340\) 0 0
\(341\) 4457.11 0.707819
\(342\) 0 0
\(343\) 7085.05 1.11533
\(344\) 0 0
\(345\) −737.884 −0.115149
\(346\) 0 0
\(347\) 3495.57 0.540783 0.270392 0.962750i \(-0.412847\pi\)
0.270392 + 0.962750i \(0.412847\pi\)
\(348\) 0 0
\(349\) −435.838 −0.0668477 −0.0334239 0.999441i \(-0.510641\pi\)
−0.0334239 + 0.999441i \(0.510641\pi\)
\(350\) 0 0
\(351\) 2576.24 0.391765
\(352\) 0 0
\(353\) 1941.73 0.292771 0.146385 0.989228i \(-0.453236\pi\)
0.146385 + 0.989228i \(0.453236\pi\)
\(354\) 0 0
\(355\) 6142.03 0.918267
\(356\) 0 0
\(357\) −1702.32 −0.252371
\(358\) 0 0
\(359\) −7188.88 −1.05687 −0.528433 0.848975i \(-0.677220\pi\)
−0.528433 + 0.848975i \(0.677220\pi\)
\(360\) 0 0
\(361\) −6231.25 −0.908478
\(362\) 0 0
\(363\) −644.014 −0.0931184
\(364\) 0 0
\(365\) −10535.4 −1.51082
\(366\) 0 0
\(367\) −1674.78 −0.238209 −0.119105 0.992882i \(-0.538002\pi\)
−0.119105 + 0.992882i \(0.538002\pi\)
\(368\) 0 0
\(369\) −5566.41 −0.785300
\(370\) 0 0
\(371\) 15149.9 2.12006
\(372\) 0 0
\(373\) 6028.31 0.836820 0.418410 0.908258i \(-0.362588\pi\)
0.418410 + 0.908258i \(0.362588\pi\)
\(374\) 0 0
\(375\) −1183.42 −0.162964
\(376\) 0 0
\(377\) 14537.7 1.98602
\(378\) 0 0
\(379\) 2065.63 0.279959 0.139979 0.990154i \(-0.455296\pi\)
0.139979 + 0.990154i \(0.455296\pi\)
\(380\) 0 0
\(381\) −888.846 −0.119520
\(382\) 0 0
\(383\) −7634.28 −1.01852 −0.509260 0.860613i \(-0.670081\pi\)
−0.509260 + 0.860613i \(0.670081\pi\)
\(384\) 0 0
\(385\) −6253.58 −0.827823
\(386\) 0 0
\(387\) 2941.18 0.386327
\(388\) 0 0
\(389\) −2814.13 −0.366792 −0.183396 0.983039i \(-0.558709\pi\)
−0.183396 + 0.983039i \(0.558709\pi\)
\(390\) 0 0
\(391\) 7445.55 0.963012
\(392\) 0 0
\(393\) −81.2511 −0.0104289
\(394\) 0 0
\(395\) −8964.08 −1.14185
\(396\) 0 0
\(397\) 8990.52 1.13658 0.568288 0.822829i \(-0.307606\pi\)
0.568288 + 0.822829i \(0.307606\pi\)
\(398\) 0 0
\(399\) 591.039 0.0741577
\(400\) 0 0
\(401\) −4829.93 −0.601484 −0.300742 0.953706i \(-0.597234\pi\)
−0.300742 + 0.953706i \(0.597234\pi\)
\(402\) 0 0
\(403\) 12325.4 1.52350
\(404\) 0 0
\(405\) 6255.05 0.767447
\(406\) 0 0
\(407\) 4555.11 0.554763
\(408\) 0 0
\(409\) −6551.78 −0.792089 −0.396045 0.918231i \(-0.629617\pi\)
−0.396045 + 0.918231i \(0.629617\pi\)
\(410\) 0 0
\(411\) −423.102 −0.0507788
\(412\) 0 0
\(413\) 17065.7 2.03329
\(414\) 0 0
\(415\) −6424.90 −0.759966
\(416\) 0 0
\(417\) 32.5446 0.00382186
\(418\) 0 0
\(419\) −10660.5 −1.24296 −0.621481 0.783429i \(-0.713469\pi\)
−0.621481 + 0.783429i \(0.713469\pi\)
\(420\) 0 0
\(421\) −8654.47 −1.00188 −0.500942 0.865481i \(-0.667013\pi\)
−0.500942 + 0.865481i \(0.667013\pi\)
\(422\) 0 0
\(423\) 1240.56 0.142596
\(424\) 0 0
\(425\) 2920.73 0.333355
\(426\) 0 0
\(427\) 16621.1 1.88373
\(428\) 0 0
\(429\) 1082.21 0.121794
\(430\) 0 0
\(431\) −4990.22 −0.557703 −0.278852 0.960334i \(-0.589954\pi\)
−0.278852 + 0.960334i \(0.589954\pi\)
\(432\) 0 0
\(433\) −7921.49 −0.879174 −0.439587 0.898200i \(-0.644875\pi\)
−0.439587 + 0.898200i \(0.644875\pi\)
\(434\) 0 0
\(435\) −1676.23 −0.184757
\(436\) 0 0
\(437\) −2585.06 −0.282975
\(438\) 0 0
\(439\) −11034.1 −1.19961 −0.599807 0.800145i \(-0.704756\pi\)
−0.599807 + 0.800145i \(0.704756\pi\)
\(440\) 0 0
\(441\) −15220.2 −1.64347
\(442\) 0 0
\(443\) −9160.66 −0.982474 −0.491237 0.871026i \(-0.663455\pi\)
−0.491237 + 0.871026i \(0.663455\pi\)
\(444\) 0 0
\(445\) 4150.80 0.442173
\(446\) 0 0
\(447\) 614.636 0.0650365
\(448\) 0 0
\(449\) −10071.1 −1.05854 −0.529271 0.848453i \(-0.677535\pi\)
−0.529271 + 0.848453i \(0.677535\pi\)
\(450\) 0 0
\(451\) −4730.20 −0.493872
\(452\) 0 0
\(453\) 1613.16 0.167314
\(454\) 0 0
\(455\) −17293.2 −1.78180
\(456\) 0 0
\(457\) −2988.54 −0.305904 −0.152952 0.988234i \(-0.548878\pi\)
−0.152952 + 0.988234i \(0.548878\pi\)
\(458\) 0 0
\(459\) 2997.33 0.304800
\(460\) 0 0
\(461\) 956.638 0.0966487 0.0483244 0.998832i \(-0.484612\pi\)
0.0483244 + 0.998832i \(0.484612\pi\)
\(462\) 0 0
\(463\) 4352.75 0.436910 0.218455 0.975847i \(-0.429898\pi\)
0.218455 + 0.975847i \(0.429898\pi\)
\(464\) 0 0
\(465\) −1421.15 −0.141730
\(466\) 0 0
\(467\) −8131.35 −0.805725 −0.402863 0.915260i \(-0.631985\pi\)
−0.402863 + 0.915260i \(0.631985\pi\)
\(468\) 0 0
\(469\) 23072.5 2.27162
\(470\) 0 0
\(471\) 1945.58 0.190335
\(472\) 0 0
\(473\) 2499.35 0.242960
\(474\) 0 0
\(475\) −1014.06 −0.0979544
\(476\) 0 0
\(477\) −13186.3 −1.26574
\(478\) 0 0
\(479\) 12912.4 1.23170 0.615849 0.787864i \(-0.288813\pi\)
0.615849 + 0.787864i \(0.288813\pi\)
\(480\) 0 0
\(481\) 12596.4 1.19407
\(482\) 0 0
\(483\) −2433.88 −0.229286
\(484\) 0 0
\(485\) −3593.92 −0.336477
\(486\) 0 0
\(487\) 11010.0 1.02446 0.512228 0.858850i \(-0.328820\pi\)
0.512228 + 0.858850i \(0.328820\pi\)
\(488\) 0 0
\(489\) 75.4990 0.00698197
\(490\) 0 0
\(491\) 14636.7 1.34531 0.672655 0.739956i \(-0.265154\pi\)
0.672655 + 0.739956i \(0.265154\pi\)
\(492\) 0 0
\(493\) 16913.9 1.54516
\(494\) 0 0
\(495\) 5443.04 0.494235
\(496\) 0 0
\(497\) 20259.2 1.82847
\(498\) 0 0
\(499\) 7549.78 0.677304 0.338652 0.940912i \(-0.390029\pi\)
0.338652 + 0.940912i \(0.390029\pi\)
\(500\) 0 0
\(501\) 1420.97 0.126715
\(502\) 0 0
\(503\) 16150.9 1.43167 0.715837 0.698267i \(-0.246045\pi\)
0.715837 + 0.698267i \(0.246045\pi\)
\(504\) 0 0
\(505\) −10229.6 −0.901410
\(506\) 0 0
\(507\) 1283.66 0.112444
\(508\) 0 0
\(509\) 19183.7 1.67053 0.835266 0.549846i \(-0.185314\pi\)
0.835266 + 0.549846i \(0.185314\pi\)
\(510\) 0 0
\(511\) −34750.6 −3.00836
\(512\) 0 0
\(513\) −1040.66 −0.0895637
\(514\) 0 0
\(515\) 8114.24 0.694284
\(516\) 0 0
\(517\) 1054.20 0.0896782
\(518\) 0 0
\(519\) −1434.13 −0.121293
\(520\) 0 0
\(521\) 6867.69 0.577503 0.288751 0.957404i \(-0.406760\pi\)
0.288751 + 0.957404i \(0.406760\pi\)
\(522\) 0 0
\(523\) −11205.2 −0.936841 −0.468420 0.883506i \(-0.655177\pi\)
−0.468420 + 0.883506i \(0.655177\pi\)
\(524\) 0 0
\(525\) −954.756 −0.0793695
\(526\) 0 0
\(527\) 14340.0 1.18531
\(528\) 0 0
\(529\) −1521.81 −0.125076
\(530\) 0 0
\(531\) −14853.8 −1.21393
\(532\) 0 0
\(533\) −13080.6 −1.06301
\(534\) 0 0
\(535\) −7729.64 −0.624639
\(536\) 0 0
\(537\) 288.218 0.0231611
\(538\) 0 0
\(539\) −12933.8 −1.03357
\(540\) 0 0
\(541\) 14471.4 1.15004 0.575021 0.818139i \(-0.304994\pi\)
0.575021 + 0.818139i \(0.304994\pi\)
\(542\) 0 0
\(543\) −674.103 −0.0532754
\(544\) 0 0
\(545\) −12145.6 −0.954603
\(546\) 0 0
\(547\) 13482.7 1.05389 0.526946 0.849899i \(-0.323337\pi\)
0.526946 + 0.849899i \(0.323337\pi\)
\(548\) 0 0
\(549\) −14466.8 −1.12464
\(550\) 0 0
\(551\) −5872.41 −0.454035
\(552\) 0 0
\(553\) −29567.6 −2.27368
\(554\) 0 0
\(555\) −1452.40 −0.111083
\(556\) 0 0
\(557\) 10194.0 0.775466 0.387733 0.921772i \(-0.373258\pi\)
0.387733 + 0.921772i \(0.373258\pi\)
\(558\) 0 0
\(559\) 6911.52 0.522945
\(560\) 0 0
\(561\) 1259.10 0.0947579
\(562\) 0 0
\(563\) −3725.21 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(564\) 0 0
\(565\) 6447.57 0.480091
\(566\) 0 0
\(567\) 20632.0 1.52815
\(568\) 0 0
\(569\) −6274.57 −0.462291 −0.231145 0.972919i \(-0.574247\pi\)
−0.231145 + 0.972919i \(0.574247\pi\)
\(570\) 0 0
\(571\) −20390.6 −1.49443 −0.747214 0.664584i \(-0.768609\pi\)
−0.747214 + 0.664584i \(0.768609\pi\)
\(572\) 0 0
\(573\) 2852.88 0.207995
\(574\) 0 0
\(575\) 4175.87 0.302862
\(576\) 0 0
\(577\) 675.385 0.0487290 0.0243645 0.999703i \(-0.492244\pi\)
0.0243645 + 0.999703i \(0.492244\pi\)
\(578\) 0 0
\(579\) −3638.09 −0.261130
\(580\) 0 0
\(581\) −21192.3 −1.51326
\(582\) 0 0
\(583\) −11205.4 −0.796019
\(584\) 0 0
\(585\) 15051.8 1.06379
\(586\) 0 0
\(587\) 5949.63 0.418343 0.209172 0.977879i \(-0.432923\pi\)
0.209172 + 0.977879i \(0.432923\pi\)
\(588\) 0 0
\(589\) −4978.78 −0.348297
\(590\) 0 0
\(591\) 2878.94 0.200379
\(592\) 0 0
\(593\) −22071.9 −1.52847 −0.764235 0.644937i \(-0.776883\pi\)
−0.764235 + 0.644937i \(0.776883\pi\)
\(594\) 0 0
\(595\) −20119.8 −1.38627
\(596\) 0 0
\(597\) 1059.19 0.0726126
\(598\) 0 0
\(599\) 9547.79 0.651272 0.325636 0.945495i \(-0.394422\pi\)
0.325636 + 0.945495i \(0.394422\pi\)
\(600\) 0 0
\(601\) −12303.7 −0.835076 −0.417538 0.908660i \(-0.637107\pi\)
−0.417538 + 0.908660i \(0.637107\pi\)
\(602\) 0 0
\(603\) −20082.0 −1.35623
\(604\) 0 0
\(605\) −7611.63 −0.511498
\(606\) 0 0
\(607\) −18252.2 −1.22049 −0.610243 0.792214i \(-0.708928\pi\)
−0.610243 + 0.792214i \(0.708928\pi\)
\(608\) 0 0
\(609\) −5528.98 −0.367891
\(610\) 0 0
\(611\) 2915.21 0.193022
\(612\) 0 0
\(613\) −13990.5 −0.921809 −0.460905 0.887450i \(-0.652475\pi\)
−0.460905 + 0.887450i \(0.652475\pi\)
\(614\) 0 0
\(615\) 1508.22 0.0988902
\(616\) 0 0
\(617\) −5747.16 −0.374995 −0.187498 0.982265i \(-0.560038\pi\)
−0.187498 + 0.982265i \(0.560038\pi\)
\(618\) 0 0
\(619\) 3913.70 0.254127 0.127064 0.991895i \(-0.459445\pi\)
0.127064 + 0.991895i \(0.459445\pi\)
\(620\) 0 0
\(621\) 4285.40 0.276920
\(622\) 0 0
\(623\) 13691.3 0.880463
\(624\) 0 0
\(625\) −8927.71 −0.571374
\(626\) 0 0
\(627\) −437.153 −0.0278440
\(628\) 0 0
\(629\) 14655.3 0.929006
\(630\) 0 0
\(631\) −26311.5 −1.65998 −0.829988 0.557781i \(-0.811653\pi\)
−0.829988 + 0.557781i \(0.811653\pi\)
\(632\) 0 0
\(633\) 2235.03 0.140339
\(634\) 0 0
\(635\) −10505.3 −0.656520
\(636\) 0 0
\(637\) −35766.1 −2.22466
\(638\) 0 0
\(639\) −17633.4 −1.09165
\(640\) 0 0
\(641\) 15538.7 0.957476 0.478738 0.877958i \(-0.341094\pi\)
0.478738 + 0.877958i \(0.341094\pi\)
\(642\) 0 0
\(643\) −6125.10 −0.375661 −0.187831 0.982201i \(-0.560146\pi\)
−0.187831 + 0.982201i \(0.560146\pi\)
\(644\) 0 0
\(645\) −796.917 −0.0486489
\(646\) 0 0
\(647\) −4635.86 −0.281692 −0.140846 0.990032i \(-0.544982\pi\)
−0.140846 + 0.990032i \(0.544982\pi\)
\(648\) 0 0
\(649\) −12622.4 −0.763440
\(650\) 0 0
\(651\) −4687.61 −0.282215
\(652\) 0 0
\(653\) −11926.6 −0.714740 −0.357370 0.933963i \(-0.616326\pi\)
−0.357370 + 0.933963i \(0.616326\pi\)
\(654\) 0 0
\(655\) −960.309 −0.0572861
\(656\) 0 0
\(657\) 30246.4 1.79608
\(658\) 0 0
\(659\) −881.384 −0.0520999 −0.0260500 0.999661i \(-0.508293\pi\)
−0.0260500 + 0.999661i \(0.508293\pi\)
\(660\) 0 0
\(661\) −19161.6 −1.12753 −0.563766 0.825935i \(-0.690648\pi\)
−0.563766 + 0.825935i \(0.690648\pi\)
\(662\) 0 0
\(663\) 3481.82 0.203956
\(664\) 0 0
\(665\) 6985.50 0.407348
\(666\) 0 0
\(667\) 24182.4 1.40382
\(668\) 0 0
\(669\) −2610.68 −0.150874
\(670\) 0 0
\(671\) −12293.6 −0.707284
\(672\) 0 0
\(673\) 20156.0 1.15447 0.577235 0.816578i \(-0.304132\pi\)
0.577235 + 0.816578i \(0.304132\pi\)
\(674\) 0 0
\(675\) 1681.07 0.0958582
\(676\) 0 0
\(677\) 5567.34 0.316056 0.158028 0.987435i \(-0.449486\pi\)
0.158028 + 0.987435i \(0.449486\pi\)
\(678\) 0 0
\(679\) −11854.4 −0.670000
\(680\) 0 0
\(681\) 3104.54 0.174694
\(682\) 0 0
\(683\) 27835.8 1.55946 0.779728 0.626118i \(-0.215357\pi\)
0.779728 + 0.626118i \(0.215357\pi\)
\(684\) 0 0
\(685\) −5000.66 −0.278927
\(686\) 0 0
\(687\) −753.013 −0.0418184
\(688\) 0 0
\(689\) −30986.6 −1.71334
\(690\) 0 0
\(691\) −17125.7 −0.942824 −0.471412 0.881913i \(-0.656255\pi\)
−0.471412 + 0.881913i \(0.656255\pi\)
\(692\) 0 0
\(693\) 17953.6 0.984129
\(694\) 0 0
\(695\) 384.646 0.0209934
\(696\) 0 0
\(697\) −15218.6 −0.827039
\(698\) 0 0
\(699\) −955.005 −0.0516761
\(700\) 0 0
\(701\) 5107.63 0.275196 0.137598 0.990488i \(-0.456062\pi\)
0.137598 + 0.990488i \(0.456062\pi\)
\(702\) 0 0
\(703\) −5088.24 −0.272983
\(704\) 0 0
\(705\) −336.131 −0.0179566
\(706\) 0 0
\(707\) −33742.0 −1.79490
\(708\) 0 0
\(709\) 11258.0 0.596335 0.298167 0.954514i \(-0.403625\pi\)
0.298167 + 0.954514i \(0.403625\pi\)
\(710\) 0 0
\(711\) 25735.3 1.35745
\(712\) 0 0
\(713\) 20502.5 1.07689
\(714\) 0 0
\(715\) 12790.7 0.669012
\(716\) 0 0
\(717\) −503.706 −0.0262360
\(718\) 0 0
\(719\) 13735.6 0.712448 0.356224 0.934401i \(-0.384064\pi\)
0.356224 + 0.934401i \(0.384064\pi\)
\(720\) 0 0
\(721\) 26764.5 1.38247
\(722\) 0 0
\(723\) −3247.21 −0.167033
\(724\) 0 0
\(725\) 9486.22 0.485944
\(726\) 0 0
\(727\) −14515.4 −0.740502 −0.370251 0.928932i \(-0.620728\pi\)
−0.370251 + 0.928932i \(0.620728\pi\)
\(728\) 0 0
\(729\) −17085.5 −0.868033
\(730\) 0 0
\(731\) 8041.22 0.406861
\(732\) 0 0
\(733\) 14218.5 0.716471 0.358236 0.933631i \(-0.383378\pi\)
0.358236 + 0.933631i \(0.383378\pi\)
\(734\) 0 0
\(735\) 4123.93 0.206957
\(736\) 0 0
\(737\) −17065.3 −0.852926
\(738\) 0 0
\(739\) −8651.36 −0.430643 −0.215322 0.976543i \(-0.569080\pi\)
−0.215322 + 0.976543i \(0.569080\pi\)
\(740\) 0 0
\(741\) −1208.87 −0.0599312
\(742\) 0 0
\(743\) 37263.7 1.83993 0.919967 0.391996i \(-0.128215\pi\)
0.919967 + 0.391996i \(0.128215\pi\)
\(744\) 0 0
\(745\) 7264.41 0.357245
\(746\) 0 0
\(747\) 18445.5 0.903460
\(748\) 0 0
\(749\) −25495.9 −1.24379
\(750\) 0 0
\(751\) 33980.7 1.65110 0.825548 0.564331i \(-0.190866\pi\)
0.825548 + 0.564331i \(0.190866\pi\)
\(752\) 0 0
\(753\) 1064.45 0.0515148
\(754\) 0 0
\(755\) 19066.0 0.919052
\(756\) 0 0
\(757\) −33247.9 −1.59632 −0.798161 0.602445i \(-0.794193\pi\)
−0.798161 + 0.602445i \(0.794193\pi\)
\(758\) 0 0
\(759\) 1800.18 0.0860901
\(760\) 0 0
\(761\) −13023.6 −0.620376 −0.310188 0.950675i \(-0.600392\pi\)
−0.310188 + 0.950675i \(0.600392\pi\)
\(762\) 0 0
\(763\) −40061.6 −1.90082
\(764\) 0 0
\(765\) 17512.0 0.827646
\(766\) 0 0
\(767\) −34905.1 −1.64322
\(768\) 0 0
\(769\) 26526.2 1.24390 0.621949 0.783057i \(-0.286341\pi\)
0.621949 + 0.783057i \(0.286341\pi\)
\(770\) 0 0
\(771\) 1330.13 0.0621316
\(772\) 0 0
\(773\) 1143.17 0.0531914 0.0265957 0.999646i \(-0.491533\pi\)
0.0265957 + 0.999646i \(0.491533\pi\)
\(774\) 0 0
\(775\) 8042.65 0.372775
\(776\) 0 0
\(777\) −4790.67 −0.221190
\(778\) 0 0
\(779\) 5283.83 0.243020
\(780\) 0 0
\(781\) −14984.4 −0.686536
\(782\) 0 0
\(783\) 9735.03 0.444319
\(784\) 0 0
\(785\) 22994.9 1.04551
\(786\) 0 0
\(787\) −41354.8 −1.87311 −0.936556 0.350519i \(-0.886005\pi\)
−0.936556 + 0.350519i \(0.886005\pi\)
\(788\) 0 0
\(789\) −4176.42 −0.188447
\(790\) 0 0
\(791\) 21267.0 0.955965
\(792\) 0 0
\(793\) −33995.8 −1.52235
\(794\) 0 0
\(795\) 3572.83 0.159390
\(796\) 0 0
\(797\) 8795.77 0.390918 0.195459 0.980712i \(-0.437380\pi\)
0.195459 + 0.980712i \(0.437380\pi\)
\(798\) 0 0
\(799\) 3391.71 0.150175
\(800\) 0 0
\(801\) −11916.7 −0.525662
\(802\) 0 0
\(803\) 25702.7 1.12955
\(804\) 0 0
\(805\) −28766.1 −1.25947
\(806\) 0 0
\(807\) −3277.77 −0.142978
\(808\) 0 0
\(809\) 2152.09 0.0935272 0.0467636 0.998906i \(-0.485109\pi\)
0.0467636 + 0.998906i \(0.485109\pi\)
\(810\) 0 0
\(811\) −24307.1 −1.05245 −0.526225 0.850346i \(-0.676393\pi\)
−0.526225 + 0.850346i \(0.676393\pi\)
\(812\) 0 0
\(813\) −3710.38 −0.160060
\(814\) 0 0
\(815\) 892.325 0.0383519
\(816\) 0 0
\(817\) −2791.87 −0.119554
\(818\) 0 0
\(819\) 49647.7 2.11823
\(820\) 0 0
\(821\) 45118.9 1.91798 0.958988 0.283446i \(-0.0914775\pi\)
0.958988 + 0.283446i \(0.0914775\pi\)
\(822\) 0 0
\(823\) −38608.9 −1.63526 −0.817631 0.575742i \(-0.804713\pi\)
−0.817631 + 0.575742i \(0.804713\pi\)
\(824\) 0 0
\(825\) 706.171 0.0298009
\(826\) 0 0
\(827\) 17908.8 0.753023 0.376512 0.926412i \(-0.377124\pi\)
0.376512 + 0.926412i \(0.377124\pi\)
\(828\) 0 0
\(829\) 42616.2 1.78543 0.892716 0.450620i \(-0.148797\pi\)
0.892716 + 0.450620i \(0.148797\pi\)
\(830\) 0 0
\(831\) 4396.12 0.183513
\(832\) 0 0
\(833\) −41612.2 −1.73082
\(834\) 0 0
\(835\) 16794.4 0.696043
\(836\) 0 0
\(837\) 8253.60 0.340844
\(838\) 0 0
\(839\) −9077.30 −0.373520 −0.186760 0.982406i \(-0.559799\pi\)
−0.186760 + 0.982406i \(0.559799\pi\)
\(840\) 0 0
\(841\) 30545.6 1.25243
\(842\) 0 0
\(843\) 2539.66 0.103761
\(844\) 0 0
\(845\) 15171.6 0.617654
\(846\) 0 0
\(847\) −25106.6 −1.01851
\(848\) 0 0
\(849\) −1570.84 −0.0634997
\(850\) 0 0
\(851\) 20953.2 0.844027
\(852\) 0 0
\(853\) −18315.0 −0.735163 −0.367581 0.929991i \(-0.619814\pi\)
−0.367581 + 0.929991i \(0.619814\pi\)
\(854\) 0 0
\(855\) −6080.09 −0.243199
\(856\) 0 0
\(857\) −13738.7 −0.547615 −0.273807 0.961785i \(-0.588283\pi\)
−0.273807 + 0.961785i \(0.588283\pi\)
\(858\) 0 0
\(859\) −9424.20 −0.374330 −0.187165 0.982328i \(-0.559930\pi\)
−0.187165 + 0.982328i \(0.559930\pi\)
\(860\) 0 0
\(861\) 4974.82 0.196912
\(862\) 0 0
\(863\) 40557.8 1.59977 0.799887 0.600151i \(-0.204893\pi\)
0.799887 + 0.600151i \(0.204893\pi\)
\(864\) 0 0
\(865\) −16950.0 −0.666263
\(866\) 0 0
\(867\) 229.191 0.00897778
\(868\) 0 0
\(869\) 21869.2 0.853698
\(870\) 0 0
\(871\) −47191.1 −1.83583
\(872\) 0 0
\(873\) 10317.9 0.400010
\(874\) 0 0
\(875\) −46135.2 −1.78246
\(876\) 0 0
\(877\) −12966.5 −0.499258 −0.249629 0.968342i \(-0.580309\pi\)
−0.249629 + 0.968342i \(0.580309\pi\)
\(878\) 0 0
\(879\) 2669.91 0.102450
\(880\) 0 0
\(881\) −15640.0 −0.598098 −0.299049 0.954238i \(-0.596669\pi\)
−0.299049 + 0.954238i \(0.596669\pi\)
\(882\) 0 0
\(883\) 10326.2 0.393548 0.196774 0.980449i \(-0.436953\pi\)
0.196774 + 0.980449i \(0.436953\pi\)
\(884\) 0 0
\(885\) 4024.65 0.152867
\(886\) 0 0
\(887\) 2173.72 0.0822846 0.0411423 0.999153i \(-0.486900\pi\)
0.0411423 + 0.999153i \(0.486900\pi\)
\(888\) 0 0
\(889\) −34651.3 −1.30727
\(890\) 0 0
\(891\) −15260.2 −0.573776
\(892\) 0 0
\(893\) −1177.58 −0.0441280
\(894\) 0 0
\(895\) 3406.46 0.127224
\(896\) 0 0
\(897\) 4978.10 0.185300
\(898\) 0 0
\(899\) 46574.9 1.72787
\(900\) 0 0
\(901\) −36051.4 −1.33301
\(902\) 0 0
\(903\) −2628.60 −0.0968706
\(904\) 0 0
\(905\) −7967.24 −0.292641
\(906\) 0 0
\(907\) −29981.6 −1.09760 −0.548800 0.835954i \(-0.684915\pi\)
−0.548800 + 0.835954i \(0.684915\pi\)
\(908\) 0 0
\(909\) 29368.6 1.07161
\(910\) 0 0
\(911\) 725.040 0.0263684 0.0131842 0.999913i \(-0.495803\pi\)
0.0131842 + 0.999913i \(0.495803\pi\)
\(912\) 0 0
\(913\) 15674.5 0.568183
\(914\) 0 0
\(915\) 3919.80 0.141623
\(916\) 0 0
\(917\) −3167.54 −0.114069
\(918\) 0 0
\(919\) 1750.94 0.0628488 0.0314244 0.999506i \(-0.489996\pi\)
0.0314244 + 0.999506i \(0.489996\pi\)
\(920\) 0 0
\(921\) 4922.39 0.176111
\(922\) 0 0
\(923\) −41436.9 −1.47769
\(924\) 0 0
\(925\) 8219.48 0.292168
\(926\) 0 0
\(927\) −23295.5 −0.825376
\(928\) 0 0
\(929\) 5500.62 0.194262 0.0971311 0.995272i \(-0.469033\pi\)
0.0971311 + 0.995272i \(0.469033\pi\)
\(930\) 0 0
\(931\) 14447.5 0.508592
\(932\) 0 0
\(933\) −7093.28 −0.248900
\(934\) 0 0
\(935\) 14881.3 0.520504
\(936\) 0 0
\(937\) −55288.1 −1.92762 −0.963811 0.266586i \(-0.914104\pi\)
−0.963811 + 0.266586i \(0.914104\pi\)
\(938\) 0 0
\(939\) 3587.37 0.124675
\(940\) 0 0
\(941\) 47729.6 1.65350 0.826748 0.562573i \(-0.190188\pi\)
0.826748 + 0.562573i \(0.190188\pi\)
\(942\) 0 0
\(943\) −21758.6 −0.751388
\(944\) 0 0
\(945\) −11580.3 −0.398631
\(946\) 0 0
\(947\) 27191.6 0.933061 0.466530 0.884505i \(-0.345504\pi\)
0.466530 + 0.884505i \(0.345504\pi\)
\(948\) 0 0
\(949\) 71076.5 2.43124
\(950\) 0 0
\(951\) −3301.17 −0.112563
\(952\) 0 0
\(953\) 2941.45 0.0999822 0.0499911 0.998750i \(-0.484081\pi\)
0.0499911 + 0.998750i \(0.484081\pi\)
\(954\) 0 0
\(955\) 33718.3 1.14251
\(956\) 0 0
\(957\) 4089.42 0.138132
\(958\) 0 0
\(959\) −16494.5 −0.555405
\(960\) 0 0
\(961\) 9696.35 0.325479
\(962\) 0 0
\(963\) 22191.3 0.742580
\(964\) 0 0
\(965\) −42998.7 −1.43438
\(966\) 0 0
\(967\) −45733.8 −1.52089 −0.760445 0.649402i \(-0.775019\pi\)
−0.760445 + 0.649402i \(0.775019\pi\)
\(968\) 0 0
\(969\) −1406.46 −0.0466276
\(970\) 0 0
\(971\) −57658.9 −1.90562 −0.952812 0.303561i \(-0.901824\pi\)
−0.952812 + 0.303561i \(0.901824\pi\)
\(972\) 0 0
\(973\) 1268.74 0.0418025
\(974\) 0 0
\(975\) 1952.80 0.0641431
\(976\) 0 0
\(977\) 46156.4 1.51144 0.755718 0.654897i \(-0.227288\pi\)
0.755718 + 0.654897i \(0.227288\pi\)
\(978\) 0 0
\(979\) −10126.5 −0.330587
\(980\) 0 0
\(981\) 34869.1 1.13485
\(982\) 0 0
\(983\) −10931.6 −0.354694 −0.177347 0.984148i \(-0.556752\pi\)
−0.177347 + 0.984148i \(0.556752\pi\)
\(984\) 0 0
\(985\) 34026.3 1.10068
\(986\) 0 0
\(987\) −1108.71 −0.0357556
\(988\) 0 0
\(989\) 11496.8 0.369644
\(990\) 0 0
\(991\) 15279.8 0.489788 0.244894 0.969550i \(-0.421247\pi\)
0.244894 + 0.969550i \(0.421247\pi\)
\(992\) 0 0
\(993\) 2958.75 0.0945549
\(994\) 0 0
\(995\) 12518.6 0.398860
\(996\) 0 0
\(997\) −7648.81 −0.242969 −0.121485 0.992593i \(-0.538766\pi\)
−0.121485 + 0.992593i \(0.538766\pi\)
\(998\) 0 0
\(999\) 8435.07 0.267141
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 752.4.a.c.1.2 3
4.3 odd 2 47.4.a.a.1.1 3
12.11 even 2 423.4.a.b.1.3 3
20.19 odd 2 1175.4.a.a.1.3 3
28.27 even 2 2303.4.a.a.1.1 3
188.187 even 2 2209.4.a.a.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
47.4.a.a.1.1 3 4.3 odd 2
423.4.a.b.1.3 3 12.11 even 2
752.4.a.c.1.2 3 1.1 even 1 trivial
1175.4.a.a.1.3 3 20.19 odd 2
2209.4.a.a.1.1 3 188.187 even 2
2303.4.a.a.1.1 3 28.27 even 2